A method of time-time analysis: The TT-transform

Abstract A novel technique of analysis of an arbitrary primary time series into a set of secondary, time-limited, local, constituent time series, the TT-transform, is presented. The time–time representation is derived from the S-transform, a method of representation of a real time series as a set of complex, time-localized spectra. When integrated over time, the S-transform becomes the Fourier transform of the primary time series. Similarly, when summed over the primary time variable, the TT-transform reverts to the primary time series. The invertibility of the TT-transform points to the possibility of filtering and signal to noise improvements in the time domain, and some insight into the localized spectra of the S-transform.

[1]  A. Grossmann,et al.  DECOMPOSITION OF HARDY FUNCTIONS INTO SQUARE INTEGRABLE WAVELETS OF CONSTANT SHAPE , 1984 .

[2]  R. P. Lowe,et al.  Local S-spectrum analysis of 1-D and 2-D data , 1997 .

[3]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[4]  R. P. Lowe,et al.  Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms , 1997 .

[5]  P. McFadden,et al.  DECOMPOSITION OF GEAR VIBRATION SIGNALS BY THE GENERALISED S TRANSFORM , 1999 .

[6]  C. Marchesi,et al.  Spectral analysis of cardiovascular time series by the S-transform , 1997, Computers in Cardiology 1997.

[7]  Peter C. Chu.  The S-Transform for Obtaining Localized Spectra , 1989 .

[8]  Stephen Theophanis,et al.  Color display of the localized spectrum , 2000 .

[9]  Olivier Rioul,et al.  Time-scale energy distributions: a general class extending wavelet transforms , 1992, IEEE Trans. Signal Process..

[10]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[11]  Dennis Gabor,et al.  Theory of communication , 1946 .

[12]  Antonia Papandreou-Suppappola,et al.  Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes , 1998, Digit. Signal Process..

[13]  Lalu Mansinha,et al.  Localization of the complex spectrum: the S transform , 1996, IEEE Trans. Signal Process..

[14]  R. G. Stockwell,et al.  S-transform analysis of gravity wave activity from a small scale network of airglow imagers , 1999 .

[15]  A. Grossmann,et al.  Cycle-octave and related transforms in seismic signal analysis , 1984 .

[16]  Khaled H. Hamed,et al.  Time-frequency analysis , 2003 .

[17]  Charles Robert Pinnegar.  The generalized S-transform and TT-transform, in one and two dimensions , 2002 .